Boundedness and concentration of random singular integrals defined by wavelet summability kernels

TL;DR

This paper investigates the boundedness and concentration properties of random singular integrals constructed from wavelet kernels with subgaussian coefficients, using probabilistic estimates to analyze their Calderón-Zygmund structure.

Contribution

It introduces a probabilistic framework to analyze the Calderón-Zygmund structure of wavelet-based kernels with random coefficients, including both smooth and Haar wavelets.

Findings

Establishes boundedness of the kernels with high probability.

Provides concentration estimates for the singular integrals.

Extends analysis to both smooth and Haar wavelet cases.

Abstract

We use Cram\'er-Chernoff type estimates in order to study the Calder\'on-Zygmund structure of the kernels $\sum_{I\in\mathcal{D}}a_I(\omega)\psi_I(x)\psi_I(y)$ where $a_I$ are subgaussian independent random variables and $\{\psi_I: I\in\mathcal{D}\}$ is a wavelet basis where $\mathcal{D}$ are the dyadic intervals in $\mathbb{R}$. We consider both, the cases of standard smooth wavelets and the case of the Haar wavelet.